Dedekind-Hasse norm implies principal ideal ring
Further information: Dedekind-Hasse norm
A Dedekind-Hasse norm on a commutative unital ring is a function from the nonzero elements of to the nonnegative integers with the property that for any elements with , it is true that either or there exists an element in the ideal such that .
Principal ideal ring
Further information: Principal ideal ring
A commutative unital ring is termed a principal ideal ring if every ideal of is principal.
Given: A commutative unital ring with a Dedekind-Hasse norm . An ideal of .
To prove: There exists such that .
Proof: If , we can take , and we are done.
So, suppose . Consider the function on the nonzero elements of . Since maps to a well-ordered set, there is an element such that for all .
Pick any . If , then or there exists with .
Note that in the latter case, we have , so , and , contradicting the assumption that has minimum norm among the nonzero elements of . Hence, we have the case . Thus, . Conversely, we clearly have , so .